Lifschitz tail for alloy-type models driven by the fractional Laplacian
| Autor: | Kamil Kaleta, Katarzyna Pietruska-Pałuba |
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| Rok vydání: | 2019 |
| Předmět: |
Unit sphere
FOS: Physical sciences Primary 60G51 60H25 Secondary 47D08 47G30 01 natural sciences Dirichlet distribution Mathematics - Spectral Theory symbols.namesake Singularity Operator (computer programming) Lattice (order) 0103 physical sciences FOS: Mathematics 0101 mathematics Spectral Theory (math.SP) Eigenvalues and eigenvectors Mathematical Physics Mathematical physics Mathematics 010102 general mathematics Probability (math.PR) Mathematical Physics (math-ph) Functional Analysis (math.FA) Mathematics - Functional Analysis Density of states symbols 010307 mathematical physics Random variable Analysis Mathematics - Probability |
| DOI: | 10.48550/arxiv.1906.03419 |
| Popis: | We establish precise asymptotics near zero of the integrated density of states for the random Schr\"{o}dinger operators $(-\Delta)^{\alpha/2} + V^{\omega}$ in $L^2(\mathbb R^d)$ for the full range of $\alpha\in(0,2]$ and a fairly large class of random nonnegative alloy-type potentials $V^{\omega}$. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit $$\lim_{s\to 0} s^{d/\alpha}\ln\ell([0,s]) = -C \left(\lambda_d^{(\alpha)}\right)^{d/\alpha},$$ with $C \in (0,\infty]$. The constant $C$ is is finite if and only if the common distribution of the lattice random variables charges $\left\{0\right\}$. In this case, the constant $C$ is expressed explicitly in terms of such a probability. In the limit formula, $\lambda_d^{(\alpha)}$ denotes the Dirichlet ground-state eigenvalue of the operator $(-\Delta)^{\alpha/2}$ in the unit ball in $\mathbb R^d.$ Comment: 23 pages |
| Databáze: | OpenAIRE |
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